http://dx.doi.org/10.1090/conm/610/12156
Asymptotic Hecke algebras and involutions
G. Lusztig
Introduction and statement of results
0.1. In [11], a Hecke algebra module structure on a vector space spanned by
the involutions in a Weyl group was defined and studied. In this paper this study is continued by relating it to the asymptotic Hecke algebra introduced in [6]. In particular we define a module over the asymptotic Hecke algebra which is spanned by the involutions in the Weyl group. We present a conjecture relating this module to equivariant vector bundles with respect to a group action on a finite set. This gives an explanation (not a proof) of a result of Kottwitz [3] in the case of classical Weyl groups, see 2.5. We also present a conjecture which realizes the module in [11] terms of an ideal in the Hecke algebra generated by a single element, see 3.4.
0.2. Let W be a Coxeter group with set of simple reflections S and with length
function l : W → N.
Let A = Z[v, v−1] where v be an indeterminate. We set u = v2. Let A be
the subring Z[u, u−1] of A. Let H (resp. H) be the free A-module (resp. free
A-module) with basis ( ˙Tw)w∈W (resp. (Tw)w∈W). We regard H (resp. H) as an
associative A-algebra (resp. A-algebra) with multiplication defined by ˙TwT˙w =
˙
Tww if l(ww) = l(w) + l(w), ( ˙Ts+ 1)( ˙Ts− u) = 0 if s ∈ S (resp. TwTw = Tww if
l(ww) = l(w) + l(w), (Ts+ 1)(Ts− u2) = 0 if s∈ S). For y, w ∈ W let Py,wbe the
polynomial defined in [2]. For w ∈ W let ˙cw = v−l(w)
y∈W ;y≤wPy,w(u) ˙Ty ∈ H,
cw = u−l(w)
y∈W ;y≤wPy,w(u2)Ty ∈ H, see [2]. Let y ≤LR w, y ∼LR w, y ∼L w
be the relations defined in [2]. We shall write , ∼ instead of ≤LR,∼LR. The
equivalence classes in W under ∼ (resp. ∼L) are called two-sided cells (resp. left
cells).
For x, y, z ∈ W we define ˙hx,y,z ∈ A, hx,y,z ∈ A by ˙cx˙cy =
z∈W ˙hx,y,z˙cz,
cxcy=
z∈Whx,y,zcz. Note that hx,y,z is obtained from ˙hx,y,z by the substitution
v→ u.
0.3. In this subsection we assume that W is a Weyl group or an (irreducible)
affine Weyl group. From the definitions we have:
(a) if ˙hx,y,z = 0 (or if hx,y,z= 0) then z x and z y.
For z∈ W there is a unique a(z) ∈ N such that ˙hx,y,z∈ va(z)Z[v−1] for all x, y∈ W 2010 Mathematics Subject Classification. Primary 20G99.
Supported in part by National Science Foundation grant DMS-0758262.
c
2014 American Mathematical Society
and ˙hx,y,z∈ v/ a(z)−1Z[v−1] for some x, y∈ W . (See [5].) Hence for z ∈ W we have
hx,y,z∈ ua(z)Z[u−1] for all x, y∈ W and hx,y,z∈ u/ a(z)−1Z[u−1] for some x, y∈ W .
For x, y, z ∈ W we have ˙hx,y,z = γx,y,z−1va(z) mod va(z)−1Z[v−1], γx,y,z−1 ∈ Z;
hence we have hx,y,z= γx,y,z−1ua(z) mod ua(z)−1Z[u−1].
(b) If x, y∈ W satisfy x y then a(x) ≥ a(y). Hence if x ∼ y then a(x) = a(y). (See [5].)
LetD be the set of distinguished involutions of W (a finite set); see [6, 2.2]). Let J be the free abelian group with basis (tw)w∈W. For x, y ∈ W we set
txty =
z∈W γx,y,z−1tz ∈ J (the sum is finite). This defines an associative ring
structure on J with unit element 1 =d∈Dtd (see [6, 2.3]).
0.4. Let∗ : W → W (or w → w∗) be an automorphism of W such that S∗= S,
∗2= 1. Let I
∗ ={w ∈ W ; w∗ = w−1}; if ∗ = 1 this is the set of involutions in W .
Let M be the freeA-module with basis (aw)w∈I∗. Following [11] for any s∈ S we
define anA-linear map Ts: M → M by
Tsaw= uaw+ (u + 1)asw if sw = ws∗> w;
Tsaw= (u2− u − 1)aw+ (u2− u)asw if sw = ws∗< w;
Tsaw= asws∗ if sw= ws∗> w;
Tsaw= (u2− 1)aw+ u2asws∗ if sw= ws∗< w.
The following result was proved in the setup of 0.3 in [11] and then in the general case in [10].
(a) These linear maps define an H-module structure on M .
Let H = A ⊗AH, M = A ⊗AM . We regard H as a subring of H and M as a
subgroup of M by ξ → 1 ⊗ ξ. Note that the H-module structure on M extends naturally to an H-module structure on M .
Let (Aw)w∈I∗ be the A-basis of M defined in [11, 0.3]. (More precisely, in
[11, 0.3] only the case where W is a Weyl group and∗ = 1 is considered in detail; the other cases are briefly mentioned in [11, 7.1]. A definition, valid in all cases is given in [10, 0.3].)
0.5. In the remainder of this section we assume that W is as in 0.3. For x ∈ W , w, w ∈ I∗ we define fx,w,w ∈ A by cxAw =
w∈I∗fx,w,wAw. The
following result is proved in 1.1:
(a) For x∈ W , w, w∈ I∗we have fx,w,w = βx,w,wv2a(w )
mod v2a(w)−1Z[v−1] where βx,w,w ∈ Z. Moreover, if βx,w,w = 0 then x ∼ w ∼ w.
Let M be the free abelian group with basis (τw)w∈I∗. For x∈ W , w ∈ I∗ we set txτw=
w∈I∗βx,w,wτw. (The last sum is finite: if βx,w,w = 0 then fx,w,w = 0
and we use the fact that cxAw is a well defined element of M .) We have the
following result.
0.6 Theorem. The bilinear pairing J× M → M defined by tx, τw→ txτw is
a (unital) J -module structure on M.
The proof is given in§1.
0.7 Notation. Let C be the field of complex numbers. For any abelian group A we set A = C⊗ A.
1. Proof of Theorem 0.6
1.1. In this section we assume that W is as in 0.3. For any x, w∈ W we have
˙cx˙cw˙cx∗−1=
(a) Hx,w,w =
y∈W ˙h(x, w, y) ˙h(y, x∗−1, w).
From the geometric description of the elements Aw in [11] one can deduce that:
(b) if x ∈ W and w, w ∈ I∗ then there exist elements Hx,w,w+ , Hx,w,w− of
N[v, v−1] such that Hx,w,w = Hx,w,w+ + Hx,w,w− and fx,w,w = Hx,w,w+ − Hx,w,w− .
(This fact has been already used in [11, 5.1] in the case where W is finite and
∗ = 1.) Let n ∈ Z, x ∈ W and w, w ∈ I∗; from (b) we deduce:
(c) If the coefficient of vn in Hx,w,w is 0 then the coefficient of vn in fx,w,w
is 0.
(d) If the coefficient of vn in H
x,w,w is 1 then the coefficient of vn in fx,w,w
is±1.
We can now prove 0.5(a). Setting a0= a(w) we have
Hx,w,w = y∈W ;wy ˙h(x, w, y) ˙h(y, x∗−1, w) = y∈W ;a(y)≤a0) ˙h(x, w, y) ˙h(y, x∗−1, w) = y∈W ;a(y)≤a0
(γx,w,y−1va(y)+ lin.comb.of va(y)−1, va(y)−2, . . . )
× (γy,x∗−1,w−1va0+ lin.comb.of va0−1, va0−2, . . . )
=
y∈W ;a(y)=a0
γx,w,y−1γy,x∗−1,w−1)v2a0+ lin.comb.of v2a0−1, v2a0−2, . . . .
Using this and (c) we deduce that
fx,w,w = βx,w,wv2a0+ lin.comb.of v2a0−1, v2a0−2, . . . )
where βx,w,w ∈ Z and that if βx,w,w = 0 then γx,w,y−1 = 0, γy,x∗−1,w−1 = 0 for
some y∈ W . For such y we have x ∼ w ∼ y−1, y∼ x∗∼ w−1, see [6, 1.9]. We see that 0.5(a) holds.
The proof above shows also:
(e) if βx,w,w = 0 then for some y ∈ W we have γx,w,y−1 = 0, γy,x∗−1,w−1 = 0.
We show:
(f) If x∈ W and w, w ∈ I∗ satisfy fx,w,w = 0 then w w and w x.
Using (c) we see that Hx,w,w = 0 hence for some y ∈ W we have ˙h(x, w, y) = 0
and ˙h(y, x−1, w)= 0. It follows that y x, y w, w y and (f) follows.
1.2. Let x, y ∈ W, w ∈ I∗. We show that (txty)τw = tx(tyτw) or equivalently
that, for any w∈ I∗,
(a)y∈Wγx,y,y−1βy,w,w =z∈I
∗βx,z,wβy,w,z
From the equality (cxcy)Aw= cx(cyAw) in M we deduce that
(b)y∈W hx,y,yfy,w,w =
z∈I∗fx,z,wfy,w,z.
Let a0 = a(w). In (b), the sum over y can be restricted to those y such that fy,w,w = 0 hence (by 1.1(f)) such that w y (hence a(y)≤ a0); the sum over z
can be restricted to those z such that fx,z,w = 0 hence (by 1.1(f)) such that w z
(hence a(z)≤ a0). Thus we have
y∈W ;a(y)≤a0
hx,y,yfy,w,w =
z∈I∗;a(z)≤a0
Using 0.5(a) this can be written as follows
y∈W ;a(y)≤a0
(γx,y,y−1v2a(y )
+ lin.comb.of va(y)−1, va(y)−2, . . . )
× (βy,w,wv2a0+ lin.comb.of v2a0−1, v2a0−2, . . . )
=
z∈I∗;a(z)≤a0
(βx,z,wv2a0+ lin.comb.of v2a0−1, v2a0−2, . . . )
× (βy,w,zv2a(z)+ lin.comb.of v2a(z)−1, v2a(z)−2, . . . )
that is, y∈W ;a(y)=a0 γx,y,y−1v2a0βy,w,wv2a0+ lin.comb.of v4a0−1, v4a0−2, = z∈I∗;a(z)=a0 βx,z,wv2a0βy,w,zv2a0)+ lin.comb.of v4a0−1, v4a0−2, . . . .
Taking the coefficient of v4a0 in both sides we obtain
y∈W ;a(y)=a0 γx,y,y−1βy,w,w = z∈I∗;a(z)=a0 βx,z,wβy,w,z.
Now, if γx,y,y−1 = 0 then a(y) = a0 and if βx,z,w = 0 then a(z) = a0. Hence we
deduce y∈W γx,y,y−1βy,w,w = z∈I∗ βx,z,wβy,w,z.
This proves (a).
1.3. Let w∈ I∗. We show that 1τw= τwor equivalently that, for any w ∈ I∗,
(a)d∈Dβd,w,w = δw,w
Let d0 be the unique element ofD contained in the left cell of w−1 (see [6, 1.10]).
If βd,w,w = 0 with d ∈ D then using 1.1(e) we can find y ∈ W such that γd,w,y−1 =
0, γy,d∗,w−1 = 0. (Note that d∗ ∈ D.) Using [6, 1.8,1.4,1.9,1.10] we deduce
γw,y−1,d= 0, γw−1,y,d∗ = 0 and y = w, y = w, d = d0, γw,y−1,d= γw−1,y,d∗ = 1.
Thusd∈Dβd,w,w = βd0,w,w and
y∈W
γd0,w,y−1γy,d∗,w−1 = γd0,w,w−1γw,d∗0,w−1δw,w = δw,w.
Thus the coefficient of v2a(w)in H
d0,w,w is δw,w. Using 1.1(c),(d) we deduce that
the coefficient of v2a(w)in f
d0,w,w is±δw,w that is, βd0,w,w =±δw,w. Thus
(b) 1τw= (w)τw
where (w) =±1. Applying 1 =d∈Dtdto both sides of (b) and using the identity
(11)τw = 1(1τw) that is 1τw = 1(1τw) we obtain (w)τw = 1((w)τw) = (w)2τw
hence (w)2= (w). Since (w) =±1 it follows that (w) = 1. This completes the
proof of (a). Theorem 0.6 is proved.
1.4. For any two-sided cell c of W let Jc(resp. Mc) be the subgroup of J (resp.
M) generated by {tx; x ∈ c} (resp. {τw; w ∈ c ∩ I∗}. Note that Jc is a subring
of J with unit element 1c =
d∈D∩cτd and J =⊕cJc (direct sum of rings). We
have M = ⊕cMc. From the last sentence in 0.5(a) we see that JcMc ⊂ Mc and
JcMc = 0 and for any two sided cells c= c. It follows that the J -module structure
1.5. For any left cell λ of W such that λ = λ∗let Jλ∩λ−1(resp. Mλ∩λ−1) be the
subgroup of J (resp. M) generated by {tx; x∈ λ∩λ−1} (resp. {τw; w∈ λ∩λ−1∩I∗}.
Note that Jλ∩λ−1is a subring of J with unit element tdwhere d is the unique element
ofD ∩ λ. Since λ = λ∗we have d = d∗. If x∈ λ ∩ λ−1, w∈ λ ∩ λ−1∩ I∗, w ∈ I∗are such that βx,w,w = 0 then w ∈ λ ∩ λ−1∩ I∗. (Indeed, as we have seen earlier, we
have γx,w,y−1 = 0, γy,x∗−1,w−1 = 0 for some y ∈ W . For such y we have x ∼Lw−1,
w ∼L y, y−1 ∼L x−1, y ∼L x∗, x∗−1 ∼L w, w−1 ∼L y−1, see [6, 1.9]. Hence
y∈ λ, y−1∈ λ w−1∈ λ, w∈ λ∗= λ, so that w∈ λ ∩ λ−1, as required.) It follows that the J -module structure onM restricts for any λ as above to a Jλ∩λ−1-module
structure onMλ∩λ−1. Now if d ∈ D−λ, w ∈ λ∩λ−1∩I∗, w ∈ I∗then βd,w,w = 0
so that tdτw= 0. (Indeed, assume that βd,w,w = 0. Then, as we have seen earlier
we have γd,w,y−1 = 0 for some y ∈ W . We then have d∼Lw−1, see [6, 1.9], hence
d ∈ λ, contradiction.) Since 1τw = τw it follows that tdτw = tw. We see that the
Jλ∩λ−1-module structure onMλ∩λ−1 is unital.
2. Γ-equivariant vector bundles
2.1. Let V ec be the category of finite dimensional vector spaces over C.
Let Γ be a finite group and let X be a finite set with a given action (a Γ-set). A Γ-equivariant C-vector bundle (or Γ-v.b.) V on X is just a collection of objects Vx∈ V ec (x ∈ X) with a given representation of Γ on ⊕x∈XVx such that
gVx= Vgx for all g∈ Γ, x ∈ X. We say that Vxis the fibre of V at x. Now X× X
is a Γ-set for the diagonal Γ-action. Let C0 be the category whose objects are the
Γ-v.b. on X× X. For V ∈ C0 let Vx,y ∈ V ec be the fibre of V at (x, y); for g ∈ Γ
letTg: Vx,y → Vgx,gy be the isomorphism given by the equivariant structure of V .
For V, V ∈ C0we define the convolution VV ∈ C0by
(VV)x,y =⊕z∈XVx,z⊗ Vz,y
for all x, y in X with the obvious Γ-equivariant structure. For V, V, V ∈ C0 we
have an obvious identification (VV)V= V(VV). Let Cδ ∈ C0 be the
Γ-v.b. given by (Cδ)x,x= C for all x∈ X and (Cδ)x,y = 0 for all x= y in X (with
the obvious Γ-equivariant structure). For V ∈ C0 we have obvious identifications CδV = V = V Cδ. Define σ : X× X → X × X by σ(x, y) = (y, x). For V ∈ C0
we set Vσ = σ∗V that is Vσ
x,y = Vy,x for all x, y in X. For V, V ∈ C0 we have an
obvious identification (VV)σ= VσVσ. Note that is compatible with direct
sums in both the V and V factor. Hence if K(C0) is the Grothendieck group ofC0
then induces an associative ring structure on K(C0) with unit element defined
by Cδ; moreover, V → Vσ induces an antiautomorphism of the ring K(C0). Thus K(C0) is an associative C-algebra with 1.
2.2. Let C be the category whose objects are pairs (U, κ) where U ∈ C0 and κ : U → U∼ σ is an isomorphism inC0 (that is a collection of isomorphisms κx,y :
Ux,y→ Uy,x for each x, y∈ X such that κgx,gyTg=Tgκx,y for all g∈ Γ, ξ, y ∈ X);
it is assumed that κy,xκx,y = 1 : Ux,y → Ux,y for all x, y∈ X.
For V ∈ C0 we define an isomorphism ζ : V ⊕ Vσ → (V ⊕ Vσ)σ= Vσ⊕ V by a⊕ b → b ⊕ a. We have (V ⊕ Vσ, ζ)∈ C and V → (V ⊕ Vσ, ζ) can be viewed as
functor Θ : C0 → C. Let K(C) be the Grothendieck group of C and let K(C) be
the subgroup of K(C) generated by the elements of the form Θ(V ) with V ∈ C0.
in [9, 11.1.5] which applies to a category with a periodic functor.) Note that if (U, κ)∈ C then (U, −κ) ∈ C and (U, κ) + (U, −κ) = 0 in ¯K(C).
For V ∈ C0, (U, κ)∈ C we define V ◦ (U, κ) ∈ C by V ◦ (U, κ) = (V UVσ, κ)
where for x, y in X,
κx,y :⊕z,z∈XVx,z⊗ Uz,z⊗ Vy,z → ⊕z,z∈XVy,z ⊗ Uz,z⊗ Vx,z
maps a⊗ b ⊗ c (in the z, z summand) to c⊗ κ(b) ⊗ a (in the z, z summand). Now
let V, V∈ C0 and (U, κ)∈ C. We have canonically
(V ⊕ V)◦ (U, κ) = V ◦ (U, κ) ⊕ V◦ (U, κ) ⊕ Θ(V UVσ).
Moreover, we have canonically V ◦ Θ(V) = Θ(VVVσ). For V, V ∈ C
0 and
(U, κ) ∈ C we have an obvious identification (VV ) ◦ (U, κ) = V◦ (V ◦ (U, κ)). For (U, κ)∈ C we have an obvious identification Cδ◦ (U, κ) = (U, κ). We see that
◦ defines a (unital) K(C0)-module structure on ¯K(C) (but not on K(C)). Hence
¯
K(C) is naturally a (unital) K(C0)-module.
2.3. Note that K(C0) has a Z-basis consisting of the the isomorphism classes
of indecomposable Γ-v.b. V on X× X (these are indexed by a Γ-orbit in X × X and an irreducible representation of the isotropy group of a point in that orbit). Moreover, ¯K(C) has a signed Z-basis consisting of the classes of (V, κ) where V is an indecomposable Γ-v.b. on X× X satisfying V ∼= Vσ and κ is defined up to a sign (so the class of (V, κ) is defined up to a sign).
2.4. Let CΓ be the category of Γ-v.b. on Γ viewed as a Γ-set under
conjuga-tion. An object Y of CΓ is a collection of objects Yg ∈ V ec (g ∈ Γ) with a given
representation of Γ on ⊕g∈ΓYg such that gYg = Yggg−1 for all g, g ∈ Γ, x ∈ X.
For Y, Y ∈ CΓ we define the convolution YY∈ CΓ by
(YY)g=⊕g1,g2∈Γ;g1g2=gYg1⊗ Y
g2
for all g ∈ Γ with the obvious Γ-equivariant structure. This defines a structure of associative ring with 1 on the Grothendieck group K(CΓ). The unit element is
given by the Γ-v.b. whose fibre at g = 1 is C and whose fibre at any other element is 0. Hence K(CΓ) is an associative C-algebra with 1; by [8, 2.2], it is commutative
and semisimple.
With X,C0,C as in 2.1, for any Y ∈ CG, we define as in [8, 2.2(h)] an object
Ψ(Y ) ∈ C0 by Ψ(Y )x,y = ⊕g∈Γ;x=gyYg (with the obvious equivariant structure).
Now Y → Ψ(Y ) defines a ring homomorphism K(CΓ) → K(C0) and a C-algebra
homomorphism K(CΓ)→ K(C0). By [8, 2.2],
(a) K(C0) is a semisimple C-algebra and the image of the homomorphism K(CΓ)→ K(C0) is exactly the centre of K(C0).
We see that the K(C0)-module structure on ¯K(C) restricts to a K(CΓ)-module
struc-ture on ¯K(C) in which the product of the class of Y ∈ CΓwith the class of (V, κ)∈ C
is the class of (V, κ)∈ C where
Vx,y =⊕g,g∈Γ,z,z∈X;x=gz,y=gzYg⊗ Vz,z⊗ Yg
that is,
(b) Vx,y =⊕g,g∈ΓYg⊗ Vg−1x,g−1y⊗ Yg
and, for x, y in X,
maps a⊗ b ⊗ c (in the g, g summand) to c⊗ κ(b) ⊗ a (in the g, g summand). It
follows also that ¯K(C) is naturally a K(CΓ)-module.
Now assume in addition that
(c) Γ is an elementary abelian 2-group
and that Y ∈ CΓ is such that for some g0 ∈ Γ, Y |Γ−{g0} is zero and dim Yg0 = 1.
Then (b) becomes
Vx,y = Yg0⊗ Vg0x,g0y⊗ Yg0
Now Yg0⊗Yg0is isomorphic to C as a representation of Γ and Vg0x,g0yis canonically
isomorphic to Vx,y. We see that (V, κ) = (V, κ). Thus Y acts as identity in
the K(CΓ)-module structure of ¯K(C). It follows that if Y is any object of CΓ
then Y acts in the K(CΓ)-module structure of ¯K(C) as multiplication by ν(Y ) =
g∈Γdim Yg. Note that ν defines a ring homomorphism K(CΓ) → Z and a
C-algebra homomorphism K(CΓ)→ C (taking 1 to 1). We see that:
(d) If Γ is as in (c) then for any ξ ∈ K(CΓ), ξ ∈ ¯K(C) we have ξξ = ν(ξ)ξ. In particular, the K(CΓ)-module ¯K(C) is ν-isotypic.
Using this and (a) we see that the first assertion in (e) below holds.
(e) If Γ is as in (c) then the K(C0)-module ¯K(C) is isotypic. Moreover dimCK(C)¯ is equal to |Γ| times the number of Γ-orbits in X.
We now prove the second assertion in (e). By 2.3, dimCK(C) is equal to n¯ Γ,X, the
number of indecomposable Γ-v.b. on Xτ X (up to isomorphism) such that V ∼= Vσ. For such V there exists a unique Γ-orbitO on X such that Vx,y= 0 implies x ∈ O
and y ∈ O. Hence nΓ,X =
OnΓ,O where O runs over the Γ-orbits in X. This
reduces the proof to the case where X is a single Γ-orbit. Let H be the isotropy group in Γ of some point in X; this is independent of the choice of point since Γ is commutative. Now any (x, y) ∈ X × X is in the same Γ-orbit as (y, x). (In-deed, we can find g ∈ Γ such that y = gx. Then (x, y) is in the same orbit as (gx, gy) = (y, g2x) = (y, x) since g2= 1.) For a given Γ-orbitO in Xτ X the
num-ber of indecomposable Γ-v.b. on X× X (up to isomorphism) with support equal to O is the number of characters of characters of the isotropy group of any point in the orbit which is H. Thus nΓ,X is equal to|H| times the number of Γ-orbits in X× X that is to |H| × |Γ/H| = |Γ|. This proves (e).
2.5. In this subsection we assume that W is an irreducible Weyl group and ∗ = 1. Let c be a two-sided cell of W such that for z ∈ c we have a(z) = 11 (if W
is of type E7) and a(z) = 11, a(z) = 26 (if W is of type E8). Let Γ be the finite
group associated to c in [8, 3.15]. For each left cell λ in c let Γλ be the subgroup of
Γ associated to λ in [8]. Let X =λ(Γ/Γλ) (λ runs over the left cells in c). Note
that Γ acts naturally on X. For any λ as above let Cλ be the Γ-v.b. on X× X
which is C at any point of form (x, x), x∈ Γ/Gλ, and is zero at all other points.
LetC0 be defined in terms of this X. The following statement was conjectured in
[8, 3.15] and proved in [1]:
(a) There exists a isomorphism φ : Jc→ K(C∼ 0) which carries the basis {tx; x∈
c} onto the canonical basis 2.3 of K(C0) and is such that for any left cell λ in c, φ(td) (where D ∩ λ = {d}) is the class of Cλ.
The isomorphism φ has the following property conjectured in [7, 10.5(b)] in a closely related situation.
(b) Let x ∈ c and let φ(tx) be the class of the indecomposable Γ-v.b. V on
Xτ X. Then φ(tx−1) is the class of ( ˇV )σ where ˇV is the dual Γ-v.b. to V .
(As R. Bezrukavnikov pointed out to me, (b) follows immediately from (a).) Next we note the following property.
(c) ˇV ∼= V for any Γ-v.b. on X× X.
It is enough to show that for any (x, y)∈ X ×X, the stabilizer of (x, y) in Γ (that is the intersection of a Γ-conjugate of Γλwith a Γ-conjugate of Γλ where λ, λare two
left cells in c) is isomorphic to a Weyl group (hence its irreducible representations are selfdual). This can be verified from the explicit description of the subgroups Γλ
in [8].
Using (b),(c) we see that φ has the following property.
(d) Let x ∈ c and let φ(tx) be the class of the indecomposable Γ-v.b. V on
Xτ X. Then φ(tx−1) is the class of Vσ.
I want to formulate a refinement of (a).
(e) Conjecture. There exists an isomorphism of abelian groups ψ :Mc→ ¯K(C)
(C is defined in terms of X) with the following properties:
-if w∈ c ∩ I∗ and φ(tw) = V (an indecomposable Γ-v.b. such that V ∼= Vσ, see
(d)) then ψ(τw) = (V, κ) for a unique choice of κ : V → V∼ σ;
-the Jc-module structure on Mc corresponds under φ and ψ to the K(C0 )-module structure on ¯K(C).
Now let Jc = C⊗ Jc, Mc = C⊗ Mc. Note that Jc is a semisimple algebra, see
[8, 1.2, 3.1(j)]. Assuming that (e) holds we deduce:
(f) If Γ is as in 2.4(c) then the Jc-moduleMc is isotypic. Moreover, dimCMc is equal to |Γ| times the number of left cells contained in c.
(Note that the number of Γ-orbits on X is equal to the number of left cells contained in c.)
Now if W is of classical type, then Γ is as in 2.4(c) and (f) gives an explanation for the known structure of the W -module obtained from M for u = 1 (a consequence of the results of Kottwitz [3]); this can be viewed as evidence for the conjecture (e). (In this case, the second assertion of (f) was already known in [4, 12.17].)
Here we use the following property which can be easily verified for any Weyl group.
(g) Let Mc (resp. Mc−c) be the A submodule of M spanned by {Ax; x
y for some y ∈ c} (resp. {Ax; x y for some y ∈ c, x /∈ c}). The decomposition
pattern of the (semisimple) Jc-moduleMcis the same as the decomposition pattern of the (semisimple) C(v)⊗AH-module C(v)⊗A(Mc/Mc−c); in particular if the first module is isotypic then so is the second module.
One can show, using results in [6, 2.8, 2.9], that this property also holds when W is replaced by an affine Weyl group and c by a finite two-sided cell in that affine Weyl group.
3. A conjectural realization of the H-module M
3.1. Let H• = Q(u)⊗AH (an algebra over Q(u)) and let M• = Q(u)⊗AM .
We regard H as a subset of H• and M as a subset of M• by ξ → 1 ⊗ ξ. The H-module structure on M extends in an obvious way to an H•-module structure on M•. Let ˆH be the vector space consisting of all formal (possibly infinite) sums
x∈WcxTx where cx ∈ Q(u). We can view H• as a subspace of ˆH in an obvious
way to a H•-module structure on ˆH. We set
X∅=
x∈W ;x∗=x
u−l(x)Tx∈ ˆH.
Let M = H•X∅be the H•-submodule of ˆH generated by X∅. In this section we will give a conjectural realization of the H•-module M•in terms of M.
We write S = {si; i ∈ I} where I is an indexing set. For any sequence
i1, i2, . . . , ik in I we write i1i2. . . ik instead of si1si2. . . sik ∈ W .
3.2. In this subsection we assume that W is of type A2,∗ = 1 and S = {s1, s2}.
We set
X∅= (−u)−3T121+ u−2T12+ u−2T21+ u−1T1+ u−1T2+ 1, X1= (1 + u)−1(T1− u)X∅ = (u− 1)(u−3T121+ u−2T12+ u−1T1), X2= (1 + u)−1(T2− u)X∅ = (u− 1)(u−3T121+ u−2T21+ u−1T2),
X121= T1X2= T2X1= (u− 1)((u−1+ u−2− u−3)T121+ u−1T12+ u−1T21).
Clearly, X∅, X1, X2, X121form a basis of M. In the H•-module M• we have a1= (u + 1)−1(T1− u)a∅, a2= (u + 1)−1(T2− u)a∅, a121= T1a2= T2a1.
We see that we have a (unique) isomorphism of H•-modules M→ M∼ • such that
X∅→ a∅, X1→ a1, X2→ a2, X121→ a121.
3.3. In this subsection we assume that W is of type A1,∗ = 1 and S = {s1}.
We set X∅ = u−1T1+ 1, X1 = (u− 1)u−1T1. Clearly, X∅, X1 form a basis of M.
In the H•-module M• we have a1 = (u + 1)−1(T1− u)a∅. We see that we have a
(unique) isomorphism of H•-modules M→ M∼ • such that X∅→ a∅, Xs→ a1. 3.4. We return to the setup in 3.1. Based on the examples in 3.2, 3.3 we state:
(a) Conjecture. There exists a unique isomorphism of H•-modules η : M→ M∼ • such that X∅→ a∅.
By 3.3, 3.2, conjecture (a) is true when W is of type A1, A2(wth∗ = 1). It can be
shown that it is also true when W is a dihedral group (any ∗) or of type A3 (any ∗).
Assuming that (a) holds we set Xw= η−1(aw) for any w∈ I∗.
3.5. We describe below the elements Xw for various w∈ I∗when W is of type
A3and∗ = 1. We write S = {s1, s2, s3} (s1s3= s3s1). X∅has been described in 3.4;
X1= (u− 1)(u−1T1+ u−2T12+ u−2T13+ u−3T121+ u−3T123+ u−3T132 + u−4T1213+ u−4T1232+ u−4T1321+ u−5T13213+ u−5T12132+ u−6T121321); X3= (u− 1)(u−1T3+ u−2T32+ u−2T13+ u−3T323+ u−3T321+ u−3T132 + u−4T3231+ u−4T3212+ u−4T1323+ u−5T13213+ u−5T32312+ u−6T121321); X2= (u− 1)(u−1T2+ u−2T21+ u−2T23+ u−3T121 + u−3T323+ u−3T213+ u−4T1213+ u−4T3231+ u−4T2132 + u−5T32312+ u−5T12132+ u−6T121321); X13= (u− 1)2(u−2T13+ u−3T132+ u−4T1321+ u−4T1323+ u−5T13213+ u−6T121321);
X121= (u− 1)(u−1T12+ u−1T21+ (u−1+ u−2− u−3)T121+ u−2T123 + u−2T213+ (u−2+ u−3− u−4)T1213+ u−3T1323+ u−3T2132 + (u−3+ u−4− u−5)T12132+ u−4T13213+ u−4T21321 + (u−4+ u−5− u−6)T121321); X323= (u− 1)(u−1T32+ u−1T23+ (u−1+ u−2− u−3)T323+ u−2T321 + u−2T213+ (u−2+ u−3− u−4)T3213+ u−3T1321+ u−3T2132 + (u−3+ u−4− u−5)T32132+ u−4T13213+ u−4T21323 + (u−4+ u−5− u−6)T121321); X2132= (u− 1)2(u−2T213+ u−3T2132+ u−4T21321 + u−4T21323+ u−4T12321+ (u−4+ u−5− u−6)T121321); X13213= (u− 1)(u−1T132+ u−1T123+ u−1T321+ (u−1+ u−2− u−3)T1321 + u−2T3213+ (u−1+ u−2− u−3)T1323+ u−2T1213+ u−2T2132 + (u−2+ u−3− u−4)T21323+ (u−2+ u−3− u−4)T21321 + (2u−2+ u−3− 2u−4)T13213+ (u−2+ 2u−3− u−4− 2u−5+ u−6)T121321); X213213= (u− 1)2(u−2T1213+ u−2T2132+ u−2T2321 + (u−2+ u−3− u−4)T21323+ (u−2+ u−3− u−4)T21321 + (u−2− u−4)T13231+ (u−2+ u−3− u−4− u−5+ u−6)T121321).
3.6. We describe below the elements Xw for various w ∈ I∗ when W is an
infinite dihedral group and ∗ = 1. We write S = {s1, s2}. X∅ has been described
in 3.4; X1= (u− 1)(u−1T1+ u−2T12+ u−3T121+ u−4T1212+ . . . ); X2= (u− 1)(u−1T2+ u−2T21+ u−3T212+ u−4T2121+ . . . ); X121= (u− 1)(u−1T12+ u−2T121+ u−3T1212+ u−4T12121+ . . . ); X212= (u− 1)(u−1T21+ u−2T212+ u−3T2121+ u−4T21212+ . . . ); X12121= (u− 1)(u−1T121+ u−2T1212+ u−3T12121+ u−4T121212+ . . . ); X21212= (u− 1)(u−1T212+ u−2T2121+ u−3T21212+ u−4T212121+ . . . ); X1212121= (u− 1)(u−1T1212+ u−2T12121+ u−3T121212+ u−4T1212121+ . . . ); X2121212= (u− 1)(u−1T2121+ u−2T21212+ u−3T212121+ u−4T2121212+ . . . ); . . .
3.7. Assume that Conjecture 3.4(a) holds for W,∗. From the examples above
we see that it is likely that the elements Xw (w ∈ I∗) are formal Z[u−1]-linear
combinations of elements Tx (x∈ W ). In particular the specializations (Xw)u−1=0
are well defined Z-linear combinations of Tx (x ∈ W ). From the example above
it appears that there is a well defined (surjective) function π : W → I∗ such that (Xw)u−1=0=
x∈π−1(w)Tx. We describe the sets π−1(w) in a few cases with∗ = 1.
If W is of type A1 we have π−1(∅) = {∅}, π−1(1) ={1}. If W is of type A2 we have π−1(∅) = {∅}, π−1(1) ={1}, π−1(2) ={2}, π−1(121) ={12, 21, 121}. If W is of type A3 we have π−1(∅) = {∅}, π−1(1) ={1}, π−1(2) ={2}, π−1(3) ={3}, π−1(13) ={13}, π−1(121) ={12, 21, 121}, π−1(323) ={32, 23, 323}, π−1(2132) ={213}, π−1(13213) ={132, 123, 321, 1321, 1323}, π−1(121321) ={1213, 2132, 2321, 21323, 21321, 13231, 121321}. If W is infinite dihedral we have
π−1(∅) = {∅}, π−1(1) ={1}, π−1(2) ={2}, π−1(121) ={12}, π−1(212) ={21},
π−1(12121) ={121}, π−1(21212) ={212}, π−1(1212121) ={1212},
π−1(2121212) ={2121}, . . .
In each of these examples π is given by the following inductive rule. We have
π(∅) = ∅. If x ∈ W is of the form x = six with i∈ I, x ∈ W , l(x) > l(x) so that
π(x) can be assumed known, then
π(x) = siπ(x) if siπ(x) = π(x)si> π(x),
π(x) = siπ(x)si if siπ(x)= π(x)si> π(x),
π(x) = π(x) if π(x)si< π(x).
In each of the examples above the following holds: if x∈ W , w = π(x) ∈ I∗, then
l(w) = l(x) + l(x−1w). We expect that these properties hold in general.
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